Morning Session Exercises

The dataset you have consists of data from 5000 hypothetical subjects on six variables, each of which has two values: cancer: whether a person has cancer. 0=no, 1=yes. try: how often a person has tried cigarettes during adolescence. 0=not often, 1=often. genes: whether a person has good or bad genes. 0=good, 1=bad. smoke: whether a person is a smoker. 0=no, 1=yes. susceptibility: whether a person is susceptible to smoking addiction. 0=no, 1=yes. fingers: whether a person has yellow-stained fingers. 0=no, 1=yes.

The model is generated using a DAG. Your job is to figure out how the arrows run. You can use two instruments for this purpose: first, you can check whether any two variables are independent, and second, you can check whether any two variables are conditionally independent, given a third.

Checking independence:

To check whether two variables X1 and X2 are independent, you type ‘ind(X1,X2)’. The output gives the contingency table for X1 and X2, the expected contingency table for X1 and X2 under independence, and a test of the null hypothesis that the variables are independent in the population. If p<.05, then the program concludes that the variables are dependent; otherwise that they are independent.

## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 3032  353
##    1 1315  300
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 5.684342e-14 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1        0        1
##    0 2942.919  442.081
##    1 1404.081  210.919
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 60.96208 
## p-value= 5.818484e-15 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 does not hold. 
## ************************************ 
##  
## 

Checking conditional independence:

To check whether two variables X1 and X2 are conditionally independent, given X3, you type ‘cind(var1=X1, var2=X2, blocker=X3)’. ‘Blocker’ is the variable you condition on. The output gives separate contingency tables for X1 and X2 for the values of X3, the expected contingency tables for X1 and X2 under conditional independence given X3, and a test of the null hypothesis that X1 and X2 are conditionally independent of X3 in the population. If p<.05, then the program concludes that the variables are conditionally dependent given the blocker; otherwise that they are independent.

## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 2517  289
##    1  573  124
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  515   64
##    1  742  176
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 1.136868e-13 
## 2 iterations: deviation 5.684342e-14 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1          0          1
##    0 2475.17556  330.82444
##    1  614.82444   82.17556
## 
## , , blocker = 1
## 
##     var2
## var1          0          1
##    0  486.17435   92.82565
##    1  770.82565  147.17435
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 45.65795 
## p-value= 1.217591e-10 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.

1. Consider the three variables ‘fingers’,’cancer’ and ‘smoke’.

a. Are ‘fingers’ and ‘cancer’ independent?

No, they are not independent.

## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 2723  336
##    1 1624  317
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 2.273737e-13 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1         0         1
##    0 2659.4946  399.5054
##    1 1687.5054  253.4946
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 29.32747 
## p-value= 6.112303e-08 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 does not hold. 
## ************************************ 
##  
## 

b. Are ‘smoke’ and ‘cancer’ conditionally independent given ‘fingers’?

According to the test, conditional independence of smoke and cancer, given fingers, does not hold.

## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 2464  278
##    1  259   58
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  568   75
##    1 1056  242
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 9.094947e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1          0          1
##    0 2440.81922  301.18078
##    1  282.18078   34.81922
## 
## , , blocker = 1
## 
##     var2
## var1          0          1
##    0  537.98660  105.01340
##    1 1086.01340  211.98660
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 32.92007 
## p-value= 7.103914e-08 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.

c. Are ‘fingers’ and ‘cancer’ conditionally independent given ‘smoke’?

According to the results, Conditional independence of fingers and cancer, given smoke, holds.

## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 2464  278
##    1  568   75
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  259   58
##    1 1056  242
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 5.684342e-14 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1          0          1
##    0 2456.05436  285.94564
##    1  575.94564   67.05436
## 
## , , blocker = 1
## 
##     var2
## var1          0          1
##    0  258.11455   58.88545
##    1 1056.88545  241.11455
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 1.285461 
## p-value= 0.5258546 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.

d. Are ‘smoke’ and ‘fingers’ conditionally independent given ‘cancer’?

According to the results, fingers and smoke, given cancer, are not conditionally independent.

## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 2464  259
##    1  568 1056
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  278   58
##    1   75  242
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1899.2721  823.7279
##    1 1132.7279  491.2721
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  181.6355  154.3645
##    1  171.3645  145.6355
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 1760.5 
## p-value= 0 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.

e. Do any additional checks you want. Which causal paths are consistent with the data for these three variables?

Conditional Independencies:

  • genes and cancer, given smoke
  • genes and cancer, given susceptibility
  • try and cancer, given smoke
  • susceptibility and try, given genes

Conditional Dependencies:

  • fingers and cancer, given try
  • smoke and cancer, given try
  • susceptibility and try, given genes
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1642  187
##    1 1390  166
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  524  128
##    1  791  172
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 2.842171e-14 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1638.2653  190.7347
##    1 1393.7347  162.2653
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  530.8854  121.1146
##    1  784.1146  178.8854
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 0.9801407 
## p-value= 0.6125833 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1724  246
##    1  408   59
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  442   69
##    1 1773  279
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1          0          1
##    0 1723.44686  246.55314
##    1  408.55314   58.44686
## 
## , , blocker = 1
## 
##     var2
## var1          0          1
##    0  441.61725   69.38275
##    1 1773.38275  278.61725
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 0.0104423 
## p-value= 0.9947925 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 2517  289
##    1  573  124
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  515   64
##    1  742  176
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 1.136868e-13 
## 2 iterations: deviation 5.684342e-14 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1          0          1
##    0 2475.17556  330.82444
##    1  614.82444   82.17556
## 
## , , blocker = 1
## 
##     var2
## var1          0          1
##    0  486.17435   92.82565
##    1  770.82565  147.17435
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 45.65795 
## p-value= 1.217591e-10 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1390  580
##    1  352  159
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  325  142
##    1 1436  616
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1383.2084  586.7916
##    1  358.7916  152.2084
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  326.4736  140.5264
##    1 1434.5264  617.4736
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 0.5673124 
## p-value= 0.7530255 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1534  436
##    1  362  105
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  295  216
##    1 1194  858
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 4.547474e-13 
## 2 iterations: deviation 2.273737e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1532.6713  437.3287
##    1  363.3287  103.6713
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  296.8705  214.1295
##    1 1192.1295  859.8705
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 0.06212018 
## p-value= 0.9694173 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1512  230
##    1 1294  467
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  317  422
##    1  262  496
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 2.273737e-13 
## 2 iterations: deviation 2.273737e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1395.3902  346.6098
##    1 1410.6098  350.3902
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  285.8257  453.1743
##    1  293.1743  464.8257
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 110.0298 
## p-value= 1.280375e-24 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 2269  537
##    1  473  106
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  145  552
##    1  172  746
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 2.273737e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 2272.9843  533.0157
##    1  469.0157  109.9843
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  136.8105  560.1895
##    1  180.1895  737.8105
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 1.286139 
## p-value= 0.5256764 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 2723  336
##    1 1624  317
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 2.273737e-13 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1         0         1
##    0 2659.4946  399.5054
##    1 1687.5054  253.4946
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 29.32747 
## p-value= 6.112303e-08 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 does not hold. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 2160  254
##    1  930  159
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  563   82
##    1  694  158
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 2129.3919  284.6081
##    1  960.6081  128.3919
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  541.5932  103.4068
##    1  715.4068  136.5932
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 21.06569 
## p-value= 2.664674e-05 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 2464  278
##    1  568   75
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  259   58
##    1 1056  242
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 5.684342e-14 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1          0          1
##    0 2456.05436  285.94564
##    1  575.94564   67.05436
## 
## , , blocker = 1
## 
##     var2
## var1          0          1
##    0  258.11455   58.88545
##    1 1056.88545  241.11455
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 1.285461 
## p-value= 0.5258546 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 2166  315
##    1 2181  338
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1         0         1
##    0 2156.9814  324.0186
##    1 2190.0186  328.9814
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 0.5732319 
## p-value= 0.4489775 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 holds. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1724  246
##    1  408   59
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  442   69
##    1 1773  279
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1          0          1
##    0 1723.44686  246.55314
##    1  408.55314   58.44686
## 
## , , blocker = 1
## 
##     var2
## var1          0          1
##    0  441.61725   69.38275
##    1 1773.38275  278.61725
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 0.0104423 
## p-value= 0.9947925 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1642  187
##    1 1390  166
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  524  128
##    1  791  172
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 2.842171e-14 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1638.2653  190.7347
##    1 1393.7347  162.2653
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  530.8854  121.1146
##    1  784.1146  178.8854
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 0.9801407 
## p-value= 0.6125833 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1539  203
##    1 1551  210
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  627  112
##    1  630  128
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1        0        1
##    0 1536.620  205.380
##    1 1553.380  207.620
## 
## , , blocker = 1
## 
##     var2
## var1        0        1
##    0  620.523  118.477
##    1  636.477  121.523
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 0.8956594 
## p-value= 0.6390135 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 3090  413
##    1 1257  240
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 4.547474e-13 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1         0         1
##    0 3045.5082  457.4918
##    1 1301.4918  195.5082
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 16.09381 
## p-value= 6.028065e-05 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 does not hold. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 2517  289
##    1  515   64
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  573  124
##    1  742  176
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 1.136868e-13 
## 2 iterations: deviation 9.094947e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1          0          1
##    0 2513.38021  292.61979
##    1  518.61979   60.38021
## 
## , , blocker = 1
## 
##     var2
## var1          0          1
##    0  567.52632  129.47368
##    1  747.47368  170.52632
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 0.789687 
## p-value= 0.6737855 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1519  196
##    1  613  109
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0 1571  217
##    1  644  131
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 1.136868e-13 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1500.3611  214.6389
##    1  631.6389   90.3611
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0 1545.2282  242.7718
##    1  669.7718  105.2282
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 16.18017 
## p-value= 0.0003065634 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1539  203
##    1  627  112
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0 1551  210
##    1  630  128
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1          0          1
##    0 1520.82709  221.17291
##    1  645.17291   93.82709
## 
## , , blocker = 1
## 
##     var2
## var1          0          1
##    0 1524.70861  236.29139
##    1  656.29139  101.70861
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 16.41624 
## p-value= 0.0002724324 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 2806  697
##    1  579  918
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1        0        1
##    0 2371.531 1131.469
##    1 1013.469  483.531
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 797.4412 
## p-value= 1.942639e-175 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 does not hold. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1512  230
##    1  317  422
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0 1294  467
##    1  262  496
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 1.136868e-13 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1284.2072  457.7928
##    1  544.7928  194.2072
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0 1087.7793  673.2207
##    1  468.2207  289.7793
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 825.4535 
## p-value= 5.689169e-180 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1552  163
##    1  344  378
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0 1254  534
##    1  235  540
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 2.273737e-13 
## 2 iterations: deviation 2.273737e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1334.2799  380.7201
##    1  561.7201  160.2799
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0 1038.7561  749.2439
##    1  450.2439  324.7561
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 858.0587 
## p-value= 4.73061e-187 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 1896  541
##    1 1489 1074
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1        0        1
##    0 1649.849  787.151
##    1 1735.151  827.849
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 225.1269 
## p-value= 6.888658e-51 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 does not hold. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1552  163
##    1 1254  534
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  344  378
##    1  235  540
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 2.273737e-13 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1373.7625  341.2375
##    1 1432.2375  355.7625
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  279.2505  442.7495
##    1  299.7495  475.2505
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 285.7444 
## p-value= 8.941064e-63 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 2742  317
##    1  643 1298
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1        0        1
##    0 2070.943  988.057
##    1 1314.057  626.943
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 1788.542 
## p-value= 0 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 does not hold. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 2269  145
##    1  537  552
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  473  172
##    1  106  746
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 2.273737e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1933.6808  480.3192
##    1  872.3192  216.6808
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  249.4689  395.5311
##    1  329.5311  522.4689
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 1499.354 
## p-value= 0 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1542  110
##    1  354  431
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0 1200  207
##    1  289  867
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1285.2655  366.7345
##    1  610.7345  174.2655
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  817.4105  589.5895
##    1  671.5895  484.4105
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 1701.273 
## p-value= 0 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1486  132
##    1  343  520
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0 1256  185
##    1  300  778
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 1.136868e-13 
## 2 iterations: deviation 2.273737e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1192.7940  425.2060
##    1  636.2060  226.7940
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  890.1135  550.8865
##    1  665.8865  412.1135
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 1755.135 
## p-value= 0 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 2414  645
##    1 1089  852
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1         0         1
##    0 2143.1354  915.8646
##    1 1359.8646  581.1354
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 290.4742 
## p-value= 3.919505e-65 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 does not hold. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1300  415
##    1  352  370
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0 1114  674
##    1  293  482
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1162.5687  552.4313
##    1  489.4313  232.5687
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  981.5513  806.4487
##    1  425.4487  349.5513
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 295.9662 
## p-value= 5.392053e-65 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 2269  537
##    1  473  106
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  145  552
##    1  172  746
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 2.273737e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 2272.9843  533.0157
##    1  469.0157  109.9843
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  136.8105  560.1895
##    1  180.1895  737.8105
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 1.286139 
## p-value= 0.5256764 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 1742  739
##    1 1761  758
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 2.273737e-13 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1         0         1
##    0 1738.1886  742.8114
##    1 1764.8114  754.1886
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 0.05540769 
## p-value= 0.8139073 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 holds. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 1715  722
##    1 1788  775
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 4.547474e-13 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1         0         1
##    0 1707.3622  729.6378
##    1 1795.6378  767.3622
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 0.2226532 
## p-value= 0.6370257 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 holds. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1390  580
##    1  325  142
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  352  159
##    1 1436  616
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 1.136868e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1386.3562  583.6438
##    1  328.6438  138.3562
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  356.4838  154.5162
##    1 1431.5162  620.4838
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 0.4000669 
## p-value= 0.8187034 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 1618  863
##    1 1441 1078
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 2.273737e-13 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1         0         1
##    0 1517.8758  963.1242
##    1 1541.1242  977.8758
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 33.82249 
## p-value= 6.037696e-09 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 does not hold. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1486  343
##    1 1256  300
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  132  520
##    1  185  778
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1481.5710  347.4290
##    1 1260.4290  295.5710
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  127.9777  524.0223
##    1  189.0223  773.9777
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 0.4146862 
## p-value= 0.8127407 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1334  636
##    1  318  149
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  284  227
##    1 1123  929
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 2.273737e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1335.4288  634.5712
##    1  316.5712  150.4288
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  280.5217  230.4783
##    1 1126.4783  925.5217
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 0.1443186 
## p-value= 0.9303827 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1534  436
##    1  362  105
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  295  216
##    1 1194  858
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 4.547474e-13 
## 2 iterations: deviation 2.273737e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1532.6713  437.3287
##    1  363.3287  103.6713
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  296.8705  214.1295
##    1 1192.1295  859.8705
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 0.06212018 
## p-value= 0.9694173 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 1970  511
##    1  467 2052
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 1.136868e-13 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1        0        1
##    0 1209.239 1271.761
##    1 1227.761 1291.239
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 1989.241 
## p-value= 0 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 does not hold. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1390  352
##    1  325 1436
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  580  159
##    1  142  616
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1        0        1
##    0 852.8490 889.1510
##    1 862.1510 898.8490
## 
## , , blocker = 1
## 
##     var2
## var1        0        1
##    0 356.4182 382.5818
##    1 365.5818 392.4182
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 1989.585 
## p-value= 0 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1534  295
##    1  362 1194
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  436  216
##    1  105  858
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 2.273737e-13 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1024.4561  804.5439
##    1  871.5439  684.4561
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  218.4099  433.5901
##    1  322.5901  640.4099
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 1907.285 
## p-value= 0 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 1715  722
##    1 1788  775
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 4.547474e-13 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1         0         1
##    0 1707.3622  729.6378
##    1 1795.6378  767.3622
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 0.2226532 
## p-value= 0.6370257 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 holds. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1552 1254
##    1  344  235
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  163  534
##    1  378  540
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1571.6916 1234.3084
##    1  324.3084  254.6916
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  233.4842  463.5158
##    1  307.5158  610.4842
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 60.84017 
## p-value= 6.147864e-14 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 3032  353
##    1 1315  300
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 5.684342e-14 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1        0        1
##    0 2942.919  442.081
##    1 1404.081  210.919
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 60.96208 
## p-value= 5.818484e-15 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 does not hold. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 2464  278
##    1  259   58
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  568   75
##    1 1056  242
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 9.094947e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1          0          1
##    0 2440.81922  301.18078
##    1  282.18078   34.81922
## 
## , , blocker = 1
## 
##     var2
## var1          0          1
##    0  537.98660  105.01340
##    1 1086.01340  211.98660
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 32.92007 
## p-value= 7.103914e-08 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 2464  278
##    1  259   58
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  568   75
##    1 1056  242
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 9.094947e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1          0          1
##    0 2440.81922  301.18078
##    1  282.18078   34.81922
## 
## , , blocker = 1
## 
##     var2
## var1          0          1
##    0  537.98660  105.01340
##    1 1086.01340  211.98660
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 32.92007 
## p-value= 7.103914e-08 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
##     var2
## var1    0    1
##    0 1829 1556
##    1  652  963
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 0 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under Independence* 
## 
##     var2
## var1        0        1
##    0 1679.637 1705.363
##    1  801.363  813.637
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 1 ) 82.01747 
## p-value= 1.348891e-19 
## 
##  
## *Conclusion* 
## Independence of Var1 and Var2 does not hold. 
## ************************************ 
##  
## 
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1534  295
##    1  436  216
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  362 1194
##    1  105  858
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 2.273737e-13 
## 2 iterations: deviation 0 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1452.2894  376.7106
##    1  517.7106  134.2894
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  288.4684 1267.5316
##    1  178.5316  784.4684
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 143.1715 
## p-value= 8.141313e-32 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, does not hold.
## 
## *Contingency Table* 
##  
## , , blocker = 0
## 
##     var2
## var1    0    1
##    0 1534  362
##    1  436  105
## 
## , , blocker = 1
## 
##     var2
## var1    0    1
##    0  295 1194
##    1  216  858
## 
## 
## *Fitting Model* 
##  
## 2 iterations: deviation 1.136868e-13 
## 2 iterations: deviation 4.547474e-13 
## 
## *Predicted frequencies under CI* 
## 
## , , blocker = 0
## 
##     var2
## var1         0         1
##    0 1532.6713  363.3287
##    1  437.3287  103.6713
## 
## , , blocker = 1
## 
##     var2
## var1         0         1
##    0  296.8705 1192.1295
##    1  214.1295  859.8705
## 
## 
##  
## *Results* 
## Likelihood-Ratio ( df = 2 ) 0.06212018 
## p-value= 0.9694173 
## 
##  
## *Conclusion* 
## Conditional independence of var1 and var2, given blocker, holds.

Afternoon Session Exercises

Conceptual

Suppose the following model is the true data-generating model of nine symptoms

Theoretically, what should the Gaussian graphical model (GGM; a network of partial correlation coefficients) of the nine observed symptoms look like? Plot or draw the expected network (use only the numbers as node labels). Tip: Note that GAD is not observed and thus (a) not in the GGM and (b) not a node we can condition on.

Suppose we only look at assignments that were graded with a 10. Are motivation and difficulty still independent? That is, in these assignments can we now predict the difficulty from motivation or vice versa? Explain your answer. Tip: if a student with a very poor motivation gets a 10, what does that tell us about the difficulty of the assignment?

Motivation and difficulty are now conditionally dependent, because they are common causes to the same consequence. If a student with very poor motivation gets a 10, this tells us that the difficulty of the assignment would be very low.

Markov Random Fields

The data frame bfiData contains the questions of the bfi (Big Five Inventory) data contained in the psych package.

Obtain the weights matrices from qgraph and bootnet by applying the getWmat function to output of both. Confirm that the results are identical (tip: the operator == tests if values in R are equal)

Qplot weight matrix:

##             A1           A2           A3           A4           A5
## A1  0.00000000 -0.279043331 -0.146342179 -0.016675805 -0.031967420
## A2 -0.27904333  0.000000000  0.275803470  0.173285222  0.108863452
## A3 -0.14634218  0.275803470  0.000000000  0.156847223  0.269593707
## A4 -0.01667580  0.173285222  0.156847223  0.000000000  0.063737085
## A5 -0.03196742  0.108863452  0.269593707  0.063737085  0.000000000
## C1  0.04029455 -0.037395321 -0.002218315 -0.068371314  0.043116099
## C2  0.05434301 -0.022258789 -0.008485116  0.187764205 -0.047483569
## C3  0.06161982  0.134280799  0.015567540 -0.036279732  0.023230906
## C4  0.10987691 -0.018992835 -0.003489887  0.002421715 -0.010696224
## C5 -0.01805194  0.035434612 -0.012961536 -0.127108242  0.017937730
## E1  0.05594849 -0.041934586  0.024394176  0.019847681  0.036466766
## E2  0.02246095 -0.038003893 -0.025894605  0.042126375 -0.019223313
## E3  0.06396481 -0.035863899  0.179257763 -0.025832709  0.177079886
## E4  0.06885956  0.004022673  0.057364661  0.134580625  0.236851213
## E5  0.06430575  0.163432325 -0.018008474 -0.015764289  0.023352938
## N1  0.04036771 -0.070769952  0.019636070  0.033789221 -0.066313167
## N2  0.05564717  0.046895017  0.026984344 -0.084303811 -0.045068233
## N3  0.03829669  0.013013859  0.007005090  0.052198539 -0.040611203
## N4 -0.03887133  0.006369286  0.013823130 -0.068996270  0.017002220
## N5 -0.05228719  0.080357098 -0.030455274  0.031196672  0.043848519
## O1  0.07613941  0.027371771 -0.012405364  0.001681125 -0.002893784
## O2  0.04409243  0.051742469 -0.002517178  0.023529803  0.038764153
## O3 -0.02102743 -0.027895099  0.048639477 -0.047559099  0.040852924
## O4 -0.08018805  0.078455271  0.007125720 -0.017702978  0.019125643
## O5  0.05146720 -0.046810133  0.023118000  0.022509445 -0.001470203
##              C1           C2           C3           C4           C5
## A1  0.040294551  0.054343013  0.061619820  0.109876914 -0.018051943
## A2 -0.037395321 -0.022258789  0.134280799 -0.018992835  0.035434612
## A3 -0.002218315 -0.008485116  0.015567540 -0.003489887 -0.012961536
## A4 -0.068371314  0.187764205 -0.036279732  0.002421715 -0.127108242
## A5  0.043116099 -0.047483569  0.023230906 -0.010696224  0.017937730
## C1  0.000000000  0.284567158  0.118521806 -0.157997932 -0.039528371
## C2  0.284567158  0.000000000  0.158467547 -0.251385463 -0.052298114
## C3  0.118521806  0.158467547  0.000000000 -0.128345587 -0.190004751
## C4 -0.157997932 -0.251385463 -0.128345587  0.000000000  0.302765187
## C5 -0.039528371 -0.052298114 -0.190004751  0.302765187  0.000000000
## E1  0.036406637  0.091828153  0.033087711  0.063361274 -0.111476299
## E2  0.023913856  0.055817642  0.034701543  0.051459159  0.094176201
## E3 -0.055307712  0.050635972 -0.023172914  0.058144645 -0.051460379
## E4  0.083691517  0.026569642 -0.020502840  0.050836910 -0.017024199
## E5  0.084726643  0.092716143  0.073344851 -0.056064749 -0.041452405
## N1 -0.025897098 -0.009565368 -0.002051581  0.095683271 -0.058468007
## N2 -0.002078596  0.012955923 -0.004605731 -0.084294346  0.110337986
## N3  0.050645425  0.001455960 -0.010552713  0.033229451  0.018954601
## N4 -0.028549895  0.065312511  0.001584436  0.043422008  0.147008894
## N5 -0.012198643  0.118505422  0.034028837  0.081150342 -0.030352418
## O1  0.029780860  0.052244427  0.014329705  0.050067065  0.017342938
## O2 -0.027117739  0.068648302  0.028673405  0.133502816  0.080295312
## O3  0.048200495  0.134709913 -0.029308714  0.118288522  0.003062263
## O4  0.114615641 -0.018620893  0.041438877  0.034253946  0.092142667
## O5 -0.033568678  0.027874381  0.068369090  0.150885045 -0.024180567
##             E1           E2           E3           E4           E5
## A1  0.05594849  0.022460951  0.063964807  0.068859558  0.064305748
## A2 -0.04193459 -0.038003893 -0.035863899  0.004022673  0.163432325
## A3  0.02439418 -0.025894605  0.179257763  0.057364661 -0.018008474
## A4  0.01984768  0.042126375 -0.025832709  0.134580625 -0.015764289
## A5  0.03646677 -0.019223313  0.177079886  0.236851213  0.023352938
## C1  0.03640664  0.023913856 -0.055307712  0.083691517  0.084726643
## C2  0.09182815  0.055817642  0.050635972  0.026569642  0.092716143
## C3  0.03308771  0.034701543 -0.023172914 -0.020502840  0.073344851
## C4  0.06336127  0.051459159  0.058144645  0.050836910 -0.056064749
## C5 -0.11147630  0.094176201 -0.051460379 -0.017024199 -0.041452405
## E1  0.00000000  0.262695652 -0.077544368 -0.235070586 -0.101845552
## E2  0.26269565  0.000000000 -0.088599010 -0.283971044 -0.143063885
## E3 -0.07754437 -0.088599010  0.000000000  0.106515893  0.147283742
## E4 -0.23507059 -0.283971044  0.106515893  0.000000000 -0.002266456
## E5 -0.10184555 -0.143063885  0.147283742 -0.002266456  0.000000000
## N1 -0.02349861 -0.029877221  0.010334721 -0.050727749  0.136429755
## N2 -0.04419810  0.057832274 -0.035643421  0.007463262  0.086245154
## N3 -0.01647635 -0.008541837  0.072370412  0.057514741 -0.046943225
## N4  0.12582556  0.038551939 -0.002479331 -0.099688865 -0.080652098
## N5 -0.11208989  0.141069386 -0.005501661  0.012573181 -0.099651449
## O1  0.04730387 -0.012981507  0.161675901  0.017213691  0.159084519
## O2  0.02711645  0.012344642  0.019642131  0.092844002  0.013008158
## O3 -0.07389055 -0.047272786  0.177562862  0.037968301  0.065147705
## O4  0.03789000  0.162811487  0.031197759 -0.022138419 -0.034137253
## O5  0.06637221  0.025408936 -0.011477943  0.146111122  0.001318067
##              N1           N2           N3            N4           N5
## A1  0.040367713  0.055647166  0.038296693 -0.0388713311 -0.052287187
## A2 -0.070769952  0.046895017  0.013013859  0.0063692861  0.080357098
## A3  0.019636070  0.026984344  0.007005090  0.0138231298 -0.030455274
## A4  0.033789221 -0.084303811  0.052198539 -0.0689962703  0.031196672
## A5 -0.066313167 -0.045068233 -0.040611203  0.0170022205  0.043848519
## C1 -0.025897098 -0.002078596  0.050645425 -0.0285498946 -0.012198643
## C2 -0.009565368  0.012955923  0.001455960  0.0653125109  0.118505422
## C3 -0.002051581 -0.004605731 -0.010552713  0.0015844359  0.034028837
## C4  0.095683271 -0.084294346  0.033229451  0.0434220077  0.081150342
## C5 -0.058468007  0.110337986  0.018954601  0.1470088941 -0.030352418
## E1 -0.023498614 -0.044198104 -0.016476354  0.1258255597 -0.112089893
## E2 -0.029877221  0.057832274 -0.008541837  0.0385519391  0.141069386
## E3  0.010334721 -0.035643421  0.072370412 -0.0024793305 -0.005501661
## E4 -0.050727749  0.007463262  0.057514741 -0.0996888652  0.012573181
## E5  0.136429755  0.086245154 -0.046943225 -0.0806520975 -0.099651449
## N1  0.000000000  0.589748200  0.208929392  0.0677400064  0.125010691
## N2  0.589748200  0.000000000  0.199546961  0.0272499944  0.024018690
## N3  0.208929392  0.199546961  0.000000000  0.3155692584  0.165349105
## N4  0.067740006  0.027249994  0.315569258  0.0000000000  0.170490574
## N5  0.125010691  0.024018690  0.165349105  0.1704905742  0.000000000
## O1 -0.004653097 -0.026904902 -0.022917713  0.0457267810 -0.066836520
## O2 -0.020558830  0.074628217 -0.027897011 -0.0219240313  0.104087581
## O3 -0.006526932  0.012978254 -0.026167476  0.0122338201  0.004860136
## O4 -0.039419118  0.043238430  0.067372291  0.0907666231  0.029367534
## O5  0.081201606 -0.065779362 -0.013644265 -0.0009882574  0.059154246
##              O1           O2           O3           O4            O5
## A1  0.076139414  0.044092425 -0.021027432 -0.080188048  0.0514672032
## A2  0.027371771  0.051742469 -0.027895099  0.078455271 -0.0468101332
## A3 -0.012405364 -0.002517178  0.048639477  0.007125720  0.0231180004
## A4  0.001681125  0.023529803 -0.047559099 -0.017702978  0.0225094451
## A5 -0.002893784  0.038764153  0.040852924  0.019125643 -0.0014702033
## C1  0.029780860 -0.027117739  0.048200495  0.114615641 -0.0335686781
## C2  0.052244427  0.068648302  0.134709913 -0.018620893  0.0278743812
## C3  0.014329705  0.028673405 -0.029308714  0.041438877  0.0683690898
## C4  0.050067065  0.133502816  0.118288522  0.034253946  0.1508850451
## C5  0.017342938  0.080295312  0.003062263  0.092142667 -0.0241805673
## E1  0.047303873  0.027116455 -0.073890547  0.037889997  0.0663722140
## E2 -0.012981507  0.012344642 -0.047272786  0.162811487  0.0254089360
## E3  0.161675901  0.019642131  0.177562862  0.031197759 -0.0114779435
## E4  0.017213691  0.092844002  0.037968301 -0.022138419  0.1461111218
## E5  0.159084519  0.013008158  0.065147705 -0.034137253  0.0013180674
## N1 -0.004653097 -0.020558830 -0.006526932 -0.039419118  0.0812016059
## N2 -0.026904902  0.074628217  0.012978254  0.043238430 -0.0657793617
## N3 -0.022917713 -0.027897011 -0.026167476  0.067372291 -0.0136442653
## N4  0.045726781 -0.021924031  0.012233820  0.090766623 -0.0009882574
## N5 -0.066836520  0.104087581  0.004860136  0.029367534  0.0591542459
## O1  0.000000000 -0.115122479  0.180775405  0.155702678 -0.1053883640
## O2 -0.115122479  0.000000000 -0.194324760 -0.002687301  0.1921171149
## O3  0.180775405 -0.194324760  0.000000000  0.144620534 -0.2013877232
## O4  0.155702678 -0.002687301  0.144620534  0.000000000 -0.1344400151
## O5 -0.105388364  0.192117115 -0.201387723 -0.134440015  0.0000000000

Bootnet weight matrix:

##             A1           A2           A3           A4           A5
## A1  0.00000000 -0.279043331 -0.146342179 -0.016675805 -0.031967420
## A2 -0.27904333  0.000000000  0.275803470  0.173285222  0.108863452
## A3 -0.14634218  0.275803470  0.000000000  0.156847223  0.269593707
## A4 -0.01667580  0.173285222  0.156847223  0.000000000  0.063737085
## A5 -0.03196742  0.108863452  0.269593707  0.063737085  0.000000000
## C1  0.04029455 -0.037395321 -0.002218315 -0.068371314  0.043116099
## C2  0.05434301 -0.022258789 -0.008485116  0.187764205 -0.047483569
## C3  0.06161982  0.134280799  0.015567540 -0.036279732  0.023230906
## C4  0.10987691 -0.018992835 -0.003489887  0.002421715 -0.010696224
## C5 -0.01805194  0.035434612 -0.012961536 -0.127108242  0.017937730
## E1  0.05594849 -0.041934586  0.024394176  0.019847681  0.036466766
## E2  0.02246095 -0.038003893 -0.025894605  0.042126375 -0.019223313
## E3  0.06396481 -0.035863899  0.179257763 -0.025832709  0.177079886
## E4  0.06885956  0.004022673  0.057364661  0.134580625  0.236851213
## E5  0.06430575  0.163432325 -0.018008474 -0.015764289  0.023352938
## N1  0.04036771 -0.070769952  0.019636070  0.033789221 -0.066313167
## N2  0.05564717  0.046895017  0.026984344 -0.084303811 -0.045068233
## N3  0.03829669  0.013013859  0.007005090  0.052198539 -0.040611203
## N4 -0.03887133  0.006369286  0.013823130 -0.068996270  0.017002220
## N5 -0.05228719  0.080357098 -0.030455274  0.031196672  0.043848519
## O1  0.07613941  0.027371771 -0.012405364  0.001681125 -0.002893784
## O2  0.04409243  0.051742469 -0.002517178  0.023529803  0.038764153
## O3 -0.02102743 -0.027895099  0.048639477 -0.047559099  0.040852924
## O4 -0.08018805  0.078455271  0.007125720 -0.017702978  0.019125643
## O5  0.05146720 -0.046810133  0.023118000  0.022509445 -0.001470203
##              C1           C2           C3           C4           C5
## A1  0.040294551  0.054343013  0.061619820  0.109876914 -0.018051943
## A2 -0.037395321 -0.022258789  0.134280799 -0.018992835  0.035434612
## A3 -0.002218315 -0.008485116  0.015567540 -0.003489887 -0.012961536
## A4 -0.068371314  0.187764205 -0.036279732  0.002421715 -0.127108242
## A5  0.043116099 -0.047483569  0.023230906 -0.010696224  0.017937730
## C1  0.000000000  0.284567158  0.118521806 -0.157997932 -0.039528371
## C2  0.284567158  0.000000000  0.158467547 -0.251385463 -0.052298114
## C3  0.118521806  0.158467547  0.000000000 -0.128345587 -0.190004751
## C4 -0.157997932 -0.251385463 -0.128345587  0.000000000  0.302765187
## C5 -0.039528371 -0.052298114 -0.190004751  0.302765187  0.000000000
## E1  0.036406637  0.091828153  0.033087711  0.063361274 -0.111476299
## E2  0.023913856  0.055817642  0.034701543  0.051459159  0.094176201
## E3 -0.055307712  0.050635972 -0.023172914  0.058144645 -0.051460379
## E4  0.083691517  0.026569642 -0.020502840  0.050836910 -0.017024199
## E5  0.084726643  0.092716143  0.073344851 -0.056064749 -0.041452405
## N1 -0.025897098 -0.009565368 -0.002051581  0.095683271 -0.058468007
## N2 -0.002078596  0.012955923 -0.004605731 -0.084294346  0.110337986
## N3  0.050645425  0.001455960 -0.010552713  0.033229451  0.018954601
## N4 -0.028549895  0.065312511  0.001584436  0.043422008  0.147008894
## N5 -0.012198643  0.118505422  0.034028837  0.081150342 -0.030352418
## O1  0.029780860  0.052244427  0.014329705  0.050067065  0.017342938
## O2 -0.027117739  0.068648302  0.028673405  0.133502816  0.080295312
## O3  0.048200495  0.134709913 -0.029308714  0.118288522  0.003062263
## O4  0.114615641 -0.018620893  0.041438877  0.034253946  0.092142667
## O5 -0.033568678  0.027874381  0.068369090  0.150885045 -0.024180567
##             E1           E2           E3           E4           E5
## A1  0.05594849  0.022460951  0.063964807  0.068859558  0.064305748
## A2 -0.04193459 -0.038003893 -0.035863899  0.004022673  0.163432325
## A3  0.02439418 -0.025894605  0.179257763  0.057364661 -0.018008474
## A4  0.01984768  0.042126375 -0.025832709  0.134580625 -0.015764289
## A5  0.03646677 -0.019223313  0.177079886  0.236851213  0.023352938
## C1  0.03640664  0.023913856 -0.055307712  0.083691517  0.084726643
## C2  0.09182815  0.055817642  0.050635972  0.026569642  0.092716143
## C3  0.03308771  0.034701543 -0.023172914 -0.020502840  0.073344851
## C4  0.06336127  0.051459159  0.058144645  0.050836910 -0.056064749
## C5 -0.11147630  0.094176201 -0.051460379 -0.017024199 -0.041452405
## E1  0.00000000  0.262695652 -0.077544368 -0.235070586 -0.101845552
## E2  0.26269565  0.000000000 -0.088599010 -0.283971044 -0.143063885
## E3 -0.07754437 -0.088599010  0.000000000  0.106515893  0.147283742
## E4 -0.23507059 -0.283971044  0.106515893  0.000000000 -0.002266456
## E5 -0.10184555 -0.143063885  0.147283742 -0.002266456  0.000000000
## N1 -0.02349861 -0.029877221  0.010334721 -0.050727749  0.136429755
## N2 -0.04419810  0.057832274 -0.035643421  0.007463262  0.086245154
## N3 -0.01647635 -0.008541837  0.072370412  0.057514741 -0.046943225
## N4  0.12582556  0.038551939 -0.002479331 -0.099688865 -0.080652098
## N5 -0.11208989  0.141069386 -0.005501661  0.012573181 -0.099651449
## O1  0.04730387 -0.012981507  0.161675901  0.017213691  0.159084519
## O2  0.02711645  0.012344642  0.019642131  0.092844002  0.013008158
## O3 -0.07389055 -0.047272786  0.177562862  0.037968301  0.065147705
## O4  0.03789000  0.162811487  0.031197759 -0.022138419 -0.034137253
## O5  0.06637221  0.025408936 -0.011477943  0.146111122  0.001318067
##              N1           N2           N3            N4           N5
## A1  0.040367713  0.055647166  0.038296693 -0.0388713311 -0.052287187
## A2 -0.070769952  0.046895017  0.013013859  0.0063692861  0.080357098
## A3  0.019636070  0.026984344  0.007005090  0.0138231298 -0.030455274
## A4  0.033789221 -0.084303811  0.052198539 -0.0689962703  0.031196672
## A5 -0.066313167 -0.045068233 -0.040611203  0.0170022205  0.043848519
## C1 -0.025897098 -0.002078596  0.050645425 -0.0285498946 -0.012198643
## C2 -0.009565368  0.012955923  0.001455960  0.0653125109  0.118505422
## C3 -0.002051581 -0.004605731 -0.010552713  0.0015844359  0.034028837
## C4  0.095683271 -0.084294346  0.033229451  0.0434220077  0.081150342
## C5 -0.058468007  0.110337986  0.018954601  0.1470088941 -0.030352418
## E1 -0.023498614 -0.044198104 -0.016476354  0.1258255597 -0.112089893
## E2 -0.029877221  0.057832274 -0.008541837  0.0385519391  0.141069386
## E3  0.010334721 -0.035643421  0.072370412 -0.0024793305 -0.005501661
## E4 -0.050727749  0.007463262  0.057514741 -0.0996888652  0.012573181
## E5  0.136429755  0.086245154 -0.046943225 -0.0806520975 -0.099651449
## N1  0.000000000  0.589748200  0.208929392  0.0677400064  0.125010691
## N2  0.589748200  0.000000000  0.199546961  0.0272499944  0.024018690
## N3  0.208929392  0.199546961  0.000000000  0.3155692584  0.165349105
## N4  0.067740006  0.027249994  0.315569258  0.0000000000  0.170490574
## N5  0.125010691  0.024018690  0.165349105  0.1704905742  0.000000000
## O1 -0.004653097 -0.026904902 -0.022917713  0.0457267810 -0.066836520
## O2 -0.020558830  0.074628217 -0.027897011 -0.0219240313  0.104087581
## O3 -0.006526932  0.012978254 -0.026167476  0.0122338201  0.004860136
## O4 -0.039419118  0.043238430  0.067372291  0.0907666231  0.029367534
## O5  0.081201606 -0.065779362 -0.013644265 -0.0009882574  0.059154246
##              O1           O2           O3           O4            O5
## A1  0.076139414  0.044092425 -0.021027432 -0.080188048  0.0514672032
## A2  0.027371771  0.051742469 -0.027895099  0.078455271 -0.0468101332
## A3 -0.012405364 -0.002517178  0.048639477  0.007125720  0.0231180004
## A4  0.001681125  0.023529803 -0.047559099 -0.017702978  0.0225094451
## A5 -0.002893784  0.038764153  0.040852924  0.019125643 -0.0014702033
## C1  0.029780860 -0.027117739  0.048200495  0.114615641 -0.0335686781
## C2  0.052244427  0.068648302  0.134709913 -0.018620893  0.0278743812
## C3  0.014329705  0.028673405 -0.029308714  0.041438877  0.0683690898
## C4  0.050067065  0.133502816  0.118288522  0.034253946  0.1508850451
## C5  0.017342938  0.080295312  0.003062263  0.092142667 -0.0241805673
## E1  0.047303873  0.027116455 -0.073890547  0.037889997  0.0663722140
## E2 -0.012981507  0.012344642 -0.047272786  0.162811487  0.0254089360
## E3  0.161675901  0.019642131  0.177562862  0.031197759 -0.0114779435
## E4  0.017213691  0.092844002  0.037968301 -0.022138419  0.1461111218
## E5  0.159084519  0.013008158  0.065147705 -0.034137253  0.0013180674
## N1 -0.004653097 -0.020558830 -0.006526932 -0.039419118  0.0812016059
## N2 -0.026904902  0.074628217  0.012978254  0.043238430 -0.0657793617
## N3 -0.022917713 -0.027897011 -0.026167476  0.067372291 -0.0136442653
## N4  0.045726781 -0.021924031  0.012233820  0.090766623 -0.0009882574
## N5 -0.066836520  0.104087581  0.004860136  0.029367534  0.0591542459
## O1  0.000000000 -0.115122479  0.180775405  0.155702678 -0.1053883640
## O2 -0.115122479  0.000000000 -0.194324760 -0.002687301  0.1921171149
## O3  0.180775405 -0.194324760  0.000000000  0.144620534 -0.2013877232
## O4  0.155702678 -0.002687301  0.144620534  0.000000000 -0.1344400151
## O5 -0.105388364  0.192117115 -0.201387723 -0.134440015  0.0000000000

What do the arguments groups and nodeNames do?

The argument groups highlights with different colors the nodes that are assigned the same given label. The argument nodeNames is used to juxtappose descriptions to the labels, and the descriptions are then shown in the legend of the plot.

In estimateNetwork, use the threshold argument to remove all edges that are not significant after applying a bonferroni correction. Plot the resulting network.

Use the default argument in estimateNetwork to estimate a partial correlation network using glasso and EBIC model selection

Set the hypertuningparameter γ to 0. Did the network change?

Compute a thresholded regularized GGM and an unregularized GGM using these new functions, and compare your results to your previous results.

Set the default to estimate a regularized Ising model using γ = 0.5 (bootnet will automatically binarize the data for you). Compare your results with the EBIC glasso network using γ = 0.5.

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